Unparticles and Emergent Mottness

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Oliver Black
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1 Unparticles and Emergent Mottness Thanks to: NSF, EFRC (DOE) Kiaran dave Charlie Kane Brandon Langley J. A. Hutasoit

2 Correlated Electron Matter

3 Correlated Electron Matter What is carrying the current?

4 fractional quantum Hall effect current-carrying excitations e = e q (q odd)

5 fractional quantum Hall effect current-carrying excitations e = e q (q odd)

6 fractional quantum Hall effect current-carrying excitations

7 fractional quantum Hall effect current-carrying excitations e =?

8 How Does Fermi Liquid Theory Breakdown? TRSB ρ at? LaSrCuO

9 How Does Fermi Liquid Theory Breakdown? TRSB ρ at? LaSrCuO

10 can the charge density be given a particle interpretation?

11 can the charge density be given a particle interpretation? does charge density= total number of electrons?

12 if not?

13 if not? charge stuff= particles + other stuff

14 if not? charge stuff= particles + other stuff unparticles!

15 counting particles

16 counting particles 1

17 counting particles 2 1

18 counting particles 2 3 1

19 counting particles

20 counting particles is there a more efficient way?

21 n = µ N(ω)dω µ IR UV

22 n = µ N(ω)dω µ IR UV can n be deduced entirely from the IR(lowenergy) scale?

23 Luttinger s `theorem

24 Luttinger s theorem n =2 k Θ(G(k,ω = 0))

25 Luttinger s theorem n =2 k Θ(G(k,ω = 0))

26 Luttinger s theorem G(E) = 1 E ε p n =2 k Θ(G(k,ω = 0)) E>ε p ε p E E<ε p counting poles (qp)

27 Luttinger s theorem G(E) = 1 E ε p n =2 k Θ(G(k,ω = 0)) zero-crossing E>ε p ε p E ε p E E<ε p counting poles (qp)

28 Luttinger s theorem G(E) = 1 E ε p n =2 k Θ(G(k,ω = 0)) zero-crossing DetG(ω =0,p)=0 ε p E>ε p E ε p E E<ε p counting poles (qp)

29 singularities of ln G n = 2i (2π) d+1 d d p 0 dξ ln GR (ξ,p) G R (ξ,p) poles+zeros (all sign changes)

30 singularities of ln G n = 2i (2π) d+1 d d p 0 dξ ln GR (ξ,p) G R (ξ,p) n =2 k Θ(G(k,ω = 0)) poles+zeros (all sign changes)

31 what are zeros?

32 what are zeros? Im G=0 µ =0

33 what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig

34 what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig = below gap+above gap

35 what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig = below gap+above gap = 0

36 what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig = below gap+above gap = 0 DetG(k,ω = 0) =0 (single band)

37 what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig = below gap+above gap = 0 DetG(k,ω = 0) =0 (single band) strongly correlated gapped systems

38 NiO insulates d 8? Mott mechanism (not Slater) t

39 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t

40 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t

41 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0

42 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone

43 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone DetReG(0,p)=0

44 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone DetReG(0,p)=0 =Mottness

45 simple problem: n=1 SU(2) U

46 simple problem: n=1 SU(2) µ U

47 simple problem: n=1 SU(2) µ U U 2

48 simple problem: n=1 SU(2) U 2 µ U U 2

49 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1 ω + U/2 + 1 ω U/2

50 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1 ω + U/2 + 1 ω U/2 =0 if ω =0

51 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1 ω + U/2 + 1 ω U/2 n =2θ(0) = 1 =0 if ω =0

52 Fermi Liquids Mott Insulators Luttinger s theorem

54 Is this famous theorem from 1960 correct?

55 A model with zeros but Luttinger fails

56 A model with zeros but Luttinger fails e t t t t t t t t

57 A model with zeros but Luttinger fails e t t t t t t t t

58 A model with zeros but Luttinger fails e t t t t t t t t no hopping=> no propagation (zeros)

59 A model with zeros but Luttinger fails N flavors e t t t t t of e- t t t no hopping=> no propagation (zeros)

60 SU(N) H = U 2 (n 1 + n N ) 2 2E 25 N =

61 SU(N) for N=3: no particle-hole symmetry

62 SU(N) for N=3: no particle-hole symmetry lim µ(t ) T 0

63 compute Green function exactly (Lehman formula) G αβ (ω) = 1 Z ab exp βk a Q ab αβ Q ab αβ = a c α bb c β a ω K b + K a + a c β bb c α a ω K a + K b do sum explicitly

64 G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N

65 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

66 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

67 < 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

68 < 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

69 a zero must exist < 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

70 Luttinger s theorem n = NΘ(2n N)

71 Luttinger s theorem n = NΘ(2n N) { 0, 1, 1/2

72 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2

73 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 2=3

74 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 2=3

75 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3

76 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd

77 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd no solution

78 does the degeneracy matter? e t t t t t t t t t =0 +

79 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω H c a ρ(0 + )

80 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω H c a ρ(0 + )

81 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + )

82 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) G ab (ω) = ωδ ab Uρ ab ω(ω U)

83 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U)

84 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U) 1/N diag(1, 1, 1, ) mixed state

85 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U) 1/N diag(1, 1, 1, ) mixed state as long as SU(N) symmetry is intact zeros in the wrong place

86 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U) 1/N diag(1, 1, 1, ) mixed state as long as SU(N) symmetry is intact zeros in the wrong place limiting procedure: lim lim t 0 ω 0

87 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U) 1/N diag(1, 1, 1, ) mixed state as long as SU(N) symmetry is intact zeros in the wrong place limiting procedure: lim lim t 0 ω 0 limits do not commute

88 Problem DetG =0 G = 1 E ε p =0??

89 Problem DetG =0 G = 1 E ε p =0?? G = 1 E ε p Σ

90 Problem DetG =0 G = 1 E ε p =0?? G = 1 E ε p Σ G=0

91 if Σ is infinite

92 if Σ is infinite lifetime of a particle vanishes

93 if Σ is infinite lifetime of a particle Σ < p vanishes

94 if Σ is infinite lifetime of a particle Σ < p vanishes no particle

95 what went wrong?

96 what went wrong? δi[g] = dωσδg

97 what went wrong? δi[g] = dωσδg if Σ

98 what went wrong? δi[g] = dωσδg if Σ integral does not exist

99 what went wrong? δi[g] = dωσδg if Σ integral does not exist No Luttinger theorem!

100 zeros and particles are incompatible

101 Luttinger s theorem

102

103 are zeros important?

104 Fermi Arcs

105 Fermi Arcs

106 Fermi Arcs Re G Changes Sign across An arc

107 Fermi Arcs Re G Changes Sign across An arc Must cross A zero line (DetG=0)!!! Fermi arcs necessarily imply zeros exist.

108 what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS)

109 what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS) seen infinities not seen zeros E F k

110 experimental data (LSCO) `Luttinger count k F 1 x FS Bi2212 zeros do not affect the particle density

111 experimental data (LSCO) `Luttinger count k F 1 x FS Bi2212 zeros do not affect the particle density each hole = a single k-state

112 where do these zeros come from?

113 counting electron states need to know: N (number of sites)

114 counting electron states x = n h /N need to know: N (number of sites)

115 counting electron states x = n h /N removal states 1 x need to know: N (number of sites)

116 counting electron states x = n h /N removal states 1 x addition states 1+x need to know: N (number of sites)

117 counting electron states 2x x = n h /N removal states 1 x addition states 1+x need to know: N (number of sites)

118 counting electron states 2x x = n h /N removal states 1 x addition states 1+x low-energy electron states 1 x +2x =1+x need to know: N (number of sites)

119 counting electron states 2x x = n h /N removal states 1 x addition states 1+x low-energy electron states 1 x +2x =1+x high energy 1 x need to know: N (number of sites)

120 spectral function (dynamics)

121 spectral function (dynamics)

122 spectral function (dynamics) density of states 1+x + α>1+x U t Harris and Lange (1967)

123 spectral function (dynamics) density of states 1+x + α>1+x α t/u U t 1 x α Harris and Lange (1967)

124 spectral function (dynamics) density of states 1+x + α>1+x α t/u U t 1 x α Harris and Lange (1967) not exhausted by counting electrons alone

125 how to count particles? some charged stuff has no particle interpretation

126

127 what is the extra stuff?

128

129 scale invariance

130 new fixed point (scale invariance)?

131 new fixed point (scale invariance)

132 new fixed point (scale invariance) unparticles (IR) (H. Georgi)

133 what is scale invariance? invariance on all length scales

134 spectral function from scale invariance A(Λω, Λ α k k)=λ α A A(ω, k)

135 spectral function from scale invariance A(Λω, Λ α k k)=λ α A A(ω, k) k A(ω, k)=ω α A f ω α k

136 what s the underlying theory?

137 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2

138 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x)

139 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x) m 2 φ 2

140 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2

141 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2

142 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2 no scale invariance

143 mass sets a scale

144

145 unparticles from a massive theory?

146 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m)

147 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2

148 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass!

149 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2

150 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!!

151 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles

152 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles

153 propagator 0 dm 2 m 2γ i p 2 m 2 + i 1 p 2 γ

154 propagator 0 dm 2 m 2γ i p 2 m 2 + i 1 p 2 γ d U 2

155 n massless particles G (E ε p ) n 2

156 n massless particles G (E ε p ) n 2 G (E ε p ) d U 2

157 n massless particles G (E ε p ) n 2 G (E ε p ) d U 2 unparticles=fractional number of massless particles

158 n massless particles G (E ε p ) n 2 G (E ε p ) d U 2 unparticles=fractional number of massless particles d U < 2 almost Luttinger liquid no pole at p_f

159 n massless particles G (E ε p ) n 2 G (E ε p ) d U 2 unparticles=fractional number of massless particles d U > 2 d U < 2 zeros almost Luttinger liquid no pole at p_f

160 n massless particles G (E ε p ) n 2 G (E ε p ) d U 2 unparticles=fractional number of massless particles d U > 2 d U < 2 zeros no simple sign change almost Luttinger liquid no pole at p_f

161 no particle interpretation mass-distribution φ U = B(m 2 )φ(m 2 )dm 2 unparticle field (non-canonical) particle field

162 d U?

163 what really is the summation over mass?

164 what really is the summation over mass? mass=energy

165 L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2

166 L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1

167 L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1 L = 0 dz 2R2 z 5+2δ 1 2 z 2 R 2 ηµν ( µ φ)( ν φ)+ φ2 2R 2

168 L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1 L = 0 dz 2R2 z 5+2δ 1 2 z 2 R 2 ηµν ( µ φ)( ν φ)+ φ2 2R 2 can be absorbed with

169 L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1 L = 0 dz 2R2 z 5+2δ 1 2 z 2 R 2 ηµν ( µ φ)( ν φ)+ φ2 2R 2 can be absorbed with ds 2 = R2 z 2 ηµν dx µ dx ν + dz 2

170 L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1 L = 0 dz 2R2 z 5+2δ 1 2 z 2 R 2 ηµν ( µ φ)( ν φ)+ φ2 2R 2 anti de Sitter can be absorbed with space! ds 2 = R2 z 2 ηµν dx µ dx ν + dz 2

171 generating functional for unparticles action on AdS 5+2δ a Φ a Φ+ Φ2 S = 1 2 d 4+2δ xdz g R 2

172 generating functional for unparticles action on AdS 5+2δ a Φ a Φ+ Φ2 S = 1 2 d 4+2δ xdz g R 2 ds 2 = L2 z 2 ηµν dx µ dx ν + dz 2 g =(R/z) 5+2δ

173 generating functional for unparticles action on AdS 5+2δ a Φ a Φ+ Φ2 S = 1 2 d 4+2δ xdz g R 2 ds 2 = L2 z 2 ηµν dx µ dx ν + dz 2 g =(R/z) 5+2δ unparticle lives in d =4+2δ δ 0

174 gauge-gravity duality (Maldacena, 1997) UV IR gravity QF T

175 gauge-gravity duality (Maldacena, 1997) UV IR gravity IR?? QF T

176 gauge-gravity duality (Maldacena, 1997) UV IR gravity IR?? QF T dg(e) dlne = β(g(e)) locality in energy

177 gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension UV IR gravity IR?? QF T dg(e) dlne = β(g(e)) locality in energy

178 Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO ))

179 Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO )) S = 1 2 d d xg zz g Φ(z,x) z Φ(z,x) z=

180 Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO )) S = 1 2 d d xg zz g Φ(z,x) z Φ(z,x) z= Φ U (x)φ U (x ) = 1 x x 2d U

181 Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO )) S = 1 2 d d xg zz g Φ(z,x) z Φ(z,x) z= Φ U (x)φ U (x ) = 1 x x 2d U d U = d 2 + d > d 2

182 Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO )) S = 1 2 d d xg zz g Φ(z,x) z Φ(z,x) z= Φ U (x)φ U (x ) = 1 x x 2d U d U = d 2 + d > d 2 scaling dimension is fixed

183 G U (p) p 2(d U d/2) d U = d 2 + d > d 2 G U (0) = 0

184 unparticle (AdS) propagator has zeros! G U (p) p 2(d U d/2) d U = d 2 + d > d 2 G U (0) = 0

185 interchanging unparticles fractional (d_u) number of massless particles

186 interchanging unparticles fractional (d_u) number of massless particles

187 interchanging unparticles fractional (d_u) number of massless particles

188 interchanging unparticles fractional (d_u) number of massless particles

189 interchanging unparticles fractional (d_u) number of massless particles e iπd U = 1, 0

190 interchanging unparticles fractional (d_u) number of massless particles e iπd U = 1, 0 fractional statistics in d=2+1

191 d U = d 2 + d > d 2 e iπd U = e iπd U time-reversal symmetry breaking from unparticle (zeros=fermi arcs) matter

192 d ln g d ln β = 4d U d>0 fermions T c unparticles with unparticle propagator BCS g tendency towards pairing (any instability which establishes a gap)

193 TRSB

194 TRSB

195 TRSB unparticles

196 TRSB unparticles particles

197 breaking of scale invariance TRSB unparticles particles

198 emergent gravity High T_c unparticles variable mass

$variable mass fractional statistics in$

199 emergent gravity High T_c unparticles variable mass fractional statistics in d=2+1

Unparticles in High T_c Superconductors

Unparticles in High T_c Superconductors Thanks to: NSF, EFRC (DOE) Kiaran dave Charlie Kane Brandon Langley J. A. Hutasoit Correlated Electron Matter Correlated Electron Matter What is carrying the current?